Polarization and Magnetization

The proper quantum mechanical treatment of the electric polarization of a dielectric material and of the orbital magnetization of a magnetic material is rather subtle. To illustrate the main complication arising in the context of DFT, let us consider a homogeneous electric field E = -V , entering the Hamiltonian via the corresponding electric potential 

Since perfectly homogeneous electric fields do not exist in nature, it seems that this lack of a Hohenberg Kohn theorem would be of little consequence for real materials and systems. However, the electronic structure of real dielectrics is very frequently calculated imder the assumption that the system is infinitely extended, and for such macroscopic dielectrics it thus appears that DFT in its usual formulation cannot be applied. 

One obvious solution is to treat the system as finite, in which case the boundary conditions to the electric field (e.g. capacitor plates) and the surface of the sample must be taken into account explicitly during the calculation. As an alternative, an additional variable can be included in the density-fimctional formalism to account for the information not contained in the charge density. Perhaps the simplest choice is the homogeneous external electric field itself Alternatively, it has been proposed to include the macroscopic polarization among the basic variables of DFT, leading to a density-and-polarization-functional theory, but it 

An alternative approach that avoids many of the problems associated with electric polarization in dielectrics is TD-DFT. In the time-dependent case, the time change of the polarization induces a current, which may be considered an ultra-nonlocal functional of the charge density, and has been successfully used as an alternative additional variable for the description of dielectric properties of both solids and molecular systems. 

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Salikhov K M, Molin Yu N, Sagdeev R Z and Buchachenko A L 1984 Spin Polarization and Magnetic Effects in Radical Reactions (Amsterdam Elsevier)... 

K. M. Salikhov.Y. N. Molin, R. Z. Sag-deev, A. L. Buchachenko, Spin Polarization and Magnetic Field Effects in Radical Reactions, Elsevier, Amsterdam, 1984. 

Here, all the extensive densities are treated as conserved quantities. This is not the general case. For example, polarization and magnetization density are not conserved. It can be shown that for nonconserved quantities, additional terms will appear on the right-hand side of Eq. 2.11. 

If this is used in the third equation of Eq. (83), the B cyclic theorem [47-61] is recovered self-consistently as follows. Without considering vacuum polarization and magnetization, the third equation of Eqs. (83) reduces to... 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

Menchero JG (1996) Spin polarization and magnetic circular dichroism in photoemission from the 2p core level of ferromagnetic Ni. Phys Rev Lett 76 3208... 

Thole BT, van der Laan G (1991) Origin of spin polarization and magnetic dichroism in corelevel photoemission. Phys Rev Lett 67 3306... 

This section is intended to provide the organic chemist with relatively simple guidelines, rational structure reactivity relationships and rules-of-thumb to predict the reactivity of biradicals and their response to changes in manageable parameters such as temperature, solvent polarity and magnetic fields. The same considerations hold, mutatis mutandis, for carbenes and nitrenes. 

The dependences of the built-in magnetic and electric fields, surface-induced magnetization and polarization on the thickness h of the freestanding tube are reported in Fig. 4.22. It is seen that the above built-in fields as well as magnetization and polarization increase monotonically as h with the tube thickness decrease. The decrease of the tube inner radius increases the built-in fields, polarization and magnetization (compare different curves plotted for different r). 

The method for analytical solution for mechanical displacements in tubes, wires and pills has been elaborated in Appendix B of Ref. [88]. Using this approach, one can derive the strain tensor components. In general case the strains contain the terms proportional to the product of flexoelectric or fiexomagnetic coefficients and polarization or magnetization gradients. They also incorporate the piezoeffect and striction coefficients as well as second powers of polarization and magnetization ... 

To study the linear FME coupling in ferroelectrics-ferromagnets we consider the model case of one-dimensional distributions of the single-component polarization and magnetization inside an ultrathin nanotube with inner R, and outer radius Ro, where the tube thickness h = Ro —Ri is very small in comparison with the average tube radius R = 0.5(Ro + Ri), see Fig. 4.29a. This simple model can be useful as it allows analytical calculations of the average quantities, which are measured by majority of conventional experimental methods. 

Fig. 4.29 (a) Sketch of one-dimensional polarization and magnetization distributions (solid curves) inside the nanotube and are corresponding extrapolation lengths, which geometrical sense is the distance on x-axes cut by the tangent line to the points x = (b) Spontaneous... 

The dependence of dielectric susceptibility and ME tunability on the magnetic field is shown in Fig. 4.34. It is seen that the effect of FME coupling between the polarization and magnetization on the tunability and dielectric susceptibility is very high. Namely in the absence of flexomagnetic effects the tunability due to the quadratic ME coupling could not exceed one percent (see Fig. 4.34b), while the FME coupling leads to the tunability of about 10-30 % (see Fig. 4.34d). 

Following the expressions in Eqs. (4.86), (4.87), (4.88) and (4.89) it is obvious that by manipulating the wire radius one can control the phase diagram (e.g. EE, AFM and FM phase transition temperatures) and corresponding spontaneous polarization and magnetization in the EuTiOs nanowires. 

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